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List the possible rational zeros. \[ n(x)=25 x^{5}+8 x^{3}-2 x^{2}+10 \] If there is more than one answer, separate them with commas. Select "None" if applicable. The possible rational zeros of \( n(x): \square \)

Ask by Lee Munoz. in the United States
Mar 15,2025

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The possible rational zeros of \( n(x) \) are \( \pm1, \pm2, \pm5, \pm10, \pm\frac{1}{5}, \pm\frac{2}{5}, \pm\frac{1}{25}, \pm\frac{2}{25} \).

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The possible rational zeros of the polynomial \( n(x) = 25x^5 + 8x^3 - 2x^2 + 10 \) can be determined using the Rational Root Theorem. The factors of the constant term (10) are ±1, ±2, ±5, ±10, and the factors of the leading coefficient (25) are ±1, ±5, ±25. To find the possible rational zeros, we take the factors of the constant term and divide them by the factors of the leading coefficient, resulting in the possibilities: ±1, ±2, ±5, ±10, ±1/5, ±2/5, ±1/25, ±2/25, ±5/25 (which simplifies to ±1/5 and ±2/5), and ±10/25 (which simplifies to ±2/5). So, the complete list of possible rational zeros is: ±1, ±2, ±5, ±10, ±1/5, ±2/5, ±1/25, ±2/25.

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